(a power series). Let UU be the interior of the set of xx such that this series converges in WW; we call UU the domain of convergence of the power series. This series defines a function from UU to WW; we are really interested in the case where UU is inhabited, in which case it is a balanced neighbourhood? of cc in VV (which is Proposition 5.3 of Bochnak–Siciak).

Let DD be any subset of VV and ff any continuous function from DD to WW. This function ff is analytic if, for every c∈Dc \in D, there is a power series as above with inhabited domain of convergence UU such that

f(x)=∑kak(x−c)k f(x) = \sum_k a_k(x - c)^k

for every xx in both DD and UU. (That ff is continuous follows automatically in many cases, including of course the finite-dimensional case.)